.5 x .5 =.25
.5^2=.25
reverse
sqrt(.25) = .5
extend that logic for all other fractions/decimal values between 0 and 1 (-1 to 0 as well if you wanna be imaginary)
When you multiply numbers bigger than 1 together they get bigger, but when you multiply numbers smaller than 1 together they get smaller (eg ½ x ½ = ¼)
So to get the square root of a number smaller than 1, the number you multiply together must be bigger than that. Eg. square root of ¼ = ½
You know that one half of one half is one quarter, right? And one quarter is smaller than one half.
Well, one half of one half is mathematically “one half times one half”, or “one half squared”. Since the result of that is one quarter, taking the square root of one quarter gets you back to one half.
I used one half as an example because it’s easier to get your head around, but this is true for all values less than one – multiplying any value less than one times another value less than one will get you a value that is smaller than *both* the original two values. Of course this also applies when you are squaring any value less than one.
Because multiplying by fractions makes things smaller, and multiplying by numbers greater than 1 makes things bigger.
If you multiply a number by 0,5 that’s the same thing as dividing it in half. It makes the product smaller.
And what happens if you divide a half in half? You get 0,25 – a quarter. That’s why the sqrt of 0,25 is 0,5.
There is no way to multiply two fractions to get a number that’s bigger than what you started with. And there’s no way to multiply two numbers larger than 1 to get a number smaller than what you started with. So if you square root a number that’s bigger than 1 you know the square root is going to be a smaller number, and if you square root a fraction you know you’re going to get a larger number.
1 is an important number for multiplication, because 1 * x = x, always.
When we multiply things by numbers bigger than 1, they get bigger. When we multiply things by numbers less than 1, they get smaller.
If B is bigger than 1, then B x is bigger than x. So B B = B^2 is bigger than B. Writing this in math: B^2 > B. So B, the square root of B^2, is less than B.
If L is less than 1, then L x is less than x. So L L = L^2 is less than L. Writing this in math: L*2 < L. So L, the square root of L^2, is less than L.
For a real world/geometric example, imagine a square. If you draw another square with sides twice as long, the area isn’t 2 times bigger, it’s 4 times bigger! Because 2^2 = 4.
Now draw a square with sides half as long. The area isn’t 1/2 times as big, it’s 1/4 times as big. Because (1/2)^2 = 1/4.
When you square things, the changes are more. If you make something bigger and square it, the square is more bigger. If you make something smaller and square, the square is more smaller.
If A is more than 1, then squaring A means multiplying A by a number that’s bigger than 1, which makes it bigger, so A squared is greater than A.
Conversely, if A is less than 1, then squaring A means multiplying it by a number that’s smaller than 1, which makes it smaller, so A squared is less than A.
Now consider that the square root is the opposite of squaring. To ask “what is the square root of A” is to ask “what number do I need to square so that the result is A”. Therefore:
from 1 and 3 follows that if A is smaller than 1, it means that the number you squared to obtain it also has to be smaller than 1. But if you square a number that’s smaller than 1, the result is smaller. Therefore A is lesser than the square root of A.
Conversely, from 2 and 3 follows that if A is greater than 1, then whatever number you squared to obtain A also has to be greater than 1. But a number that’s greater than 1 squared is greater than itself, therefore A is greater than its square root.
If you multiply some random number with any number between 0 and 1, you are basically making the number smaller. So the square of a number between 0 and 1 is smaller than itself, as it becomes smaller when multiplied by itself.
But above 1, multiplication leads to increase in the value. So squares are always larger than the roots.
Think of multiplying by numbers below 1 but above 0 as multiplying by some number and then dividing by a larger number. So the result is always smaller.
Comments
Because a square root is literally the opposite function of squaring a value.
For anything below 1: taking a fraction of a fraction makes a number smaller. For anything above 1: multiplying a multiple makes it bigger.
So it stands to reason that the inverse happens with the square root.
Because multiplying a number by something less than 1 makes the number smaller.
0.5 * 0.5 = 0.25
1/2 * 1/2 = 1/4, so the square root of a 1/4 is 1/2.
2 * 2 = 4, so the square root of 4 is 2.
Not sure what else to say except that is how the math works.
.5 x .5 =.25
.5^2=.25
reverse
sqrt(.25) = .5
extend that logic for all other fractions/decimal values between 0 and 1 (-1 to 0 as well if you wanna be imaginary)
When you multiply numbers bigger than 1 together they get bigger, but when you multiply numbers smaller than 1 together they get smaller (eg ½ x ½ = ¼)
So to get the square root of a number smaller than 1, the number you multiply together must be bigger than that. Eg. square root of ¼ = ½
You know that one half of one half is one quarter, right? And one quarter is smaller than one half.
Well, one half of one half is mathematically “one half times one half”, or “one half squared”. Since the result of that is one quarter, taking the square root of one quarter gets you back to one half.
I used one half as an example because it’s easier to get your head around, but this is true for all values less than one – multiplying any value less than one times another value less than one will get you a value that is smaller than *both* the original two values. Of course this also applies when you are squaring any value less than one.
Because multiplying by fractions makes things smaller, and multiplying by numbers greater than 1 makes things bigger.
If you multiply a number by 0,5 that’s the same thing as dividing it in half. It makes the product smaller.
And what happens if you divide a half in half? You get 0,25 – a quarter. That’s why the sqrt of 0,25 is 0,5.
There is no way to multiply two fractions to get a number that’s bigger than what you started with. And there’s no way to multiply two numbers larger than 1 to get a number smaller than what you started with. So if you square root a number that’s bigger than 1 you know the square root is going to be a smaller number, and if you square root a fraction you know you’re going to get a larger number.
1 is an important number for multiplication, because 1 * x = x, always.
When we multiply things by numbers bigger than 1, they get bigger. When we multiply things by numbers less than 1, they get smaller.
If B is bigger than 1, then B x is bigger than x. So B B = B^2 is bigger than B. Writing this in math: B^2 > B. So B, the square root of B^2, is less than B.
If L is less than 1, then L x is less than x. So L L = L^2 is less than L. Writing this in math: L*2 < L. So L, the square root of L^2, is less than L.
For a real world/geometric example, imagine a square. If you draw another square with sides twice as long, the area isn’t 2 times bigger, it’s 4 times bigger! Because 2^2 = 4.
Now draw a square with sides half as long. The area isn’t 1/2 times as big, it’s 1/4 times as big. Because (1/2)^2 = 1/4.
When you square things, the changes are more. If you make something bigger and square it, the square is more bigger. If you make something smaller and square, the square is more smaller.
Consider any positive number A.
If A is more than 1, then squaring A means multiplying A by a number that’s bigger than 1, which makes it bigger, so A squared is greater than A.
Conversely, if A is less than 1, then squaring A means multiplying it by a number that’s smaller than 1, which makes it smaller, so A squared is less than A.
Now consider that the square root is the opposite of squaring. To ask “what is the square root of A” is to ask “what number do I need to square so that the result is A”. Therefore:
from 1 and 3 follows that if A is smaller than 1, it means that the number you squared to obtain it also has to be smaller than 1. But if you square a number that’s smaller than 1, the result is smaller. Therefore A is lesser than the square root of A.
Conversely, from 2 and 3 follows that if A is greater than 1, then whatever number you squared to obtain A also has to be greater than 1. But a number that’s greater than 1 squared is greater than itself, therefore A is greater than its square root.
There you have it.
If you multiply some random number with any number between 0 and 1, you are basically making the number smaller. So the square of a number between 0 and 1 is smaller than itself, as it becomes smaller when multiplied by itself.
But above 1, multiplication leads to increase in the value. So squares are always larger than the roots.
Think of multiplying by numbers below 1 but above 0 as multiplying by some number and then dividing by a larger number. So the result is always smaller.